In the present paper, a polynomial invariant [Formula: see text] for virtual knots is constructed by 0-smoothing at all real crossings and using the polynomial invariant [Formula: see text] for flat virtual knots. We further discuss some properties of the polynomial invariant [Formula: see text]. Finally, we construct a family of diagrams for virtual knots which can be distinguished from each other by invariant [Formula: see text].
We consider the representation [Formula: see text] which extends the Lawrence-Bigelow-Krammer representation from the braid group [Formula: see text] to the virtual braid group [Formula: see text] on 3 strands. After specializing parameters [Formula: see text], [Formula: see text] to non-zero complex numbers, we study the irreducibility of [Formula: see text]. We prove that [Formula: see text] is irreducible for all non-zero complex numbers [Formula: see text], [Formula: see text] except for the case [Formula: see text].
In this paper, we give Bennequin–Plamenevskaya–Shumakovitch-type lower bounds for the concordance invariant [Formula: see text] introduced by Kronheimer and Mrowka. The proof is a consequence of computations for torus knots and the cobordism inequality of [Formula: see text] due to Gong, combined with well-known arguments used for slice-torus invariants.
For a knot [Formula: see text], let [Formula: see text] denote the set of nontrivial knot types represented by simple closed curves on a minimal genus Seifert surface of [Formula: see text]. We study the relation [Formula: see text] and its symmetric part, which leads to the notion of amicable knots: Knots [Formula: see text] and [Formula: see text] are called amicable if each is represented by a simple closed curve on a minimal genus Seifert surface of the other. A classical result of Lyon implies that the family of torus knots is universal for this realization problem: For every nontrivial knot type [Formula: see text], there exists a torus knot [Formula: see text] such that [Formula: see text]. In contrast, one of the main results of this paper is that no single knot is universal: For every knot [Formula: see text], there exists a knot [Formula: see text] such that [Formula: see text]. We also study explicit examples, keeping track of chirality throughout. Writing [Formula: see text] and [Formula: see text] for the right-handed positive torus knots, we show that [Formula: see text] and [Formula: see text] are amicable, whereas [Formula: see text] and the figure-eight knot [Formula: see text] are not. We also describe the hosting sets of both chiralities of the trefoil in terms of primitive slope classes on their once-punctured torus fibers.
We construct a new polynomial invariant of flat virtual knots by using the MD-polynomial introduced in an earlier work of the author. In defining the MD-polynomial, we assign a monomial m(c) to each crossing c of a virtual knot diagram D. By modifying m(c) suitably, we define a polynomial for flat virtual knots. We show that FMD is invariant under the flat Reidemeister moves, and hence is a flat virtual knot invariant. Furthermore, explicit computations demonstrate that the new invariant distinguishes flat virtual knots that are invisible to the Kauffman-Richter polynomial P-D. We show that if FMD = FMD ' then P-D = P-D '. Moreover, we generalize the polynomial FMD to FMD(n) for each natural number n.
This note is in two parts. The first part contains proofs of four elementary criteria for two permutations in the symmetric group Sn to be conjugate in S n . Most readers know the first criterion but perhaps fewer are familiar with the remaining three. The second part contains a discussion of the problem of deciding whether two ordered pairs of permutations are conjugate by an element in S n . In sharp contrast to the situation in part one, here no elementary criteria are known. However, pairs of permutations give rise to dessins d’enfants and two ordered pairs of permutations are conjugate precisely when the corresponding dessins are isomorphic. We give an expository discussion, with examples and references but no proofs, of some of this material; we hope our discussion will help readers who are encountering this known material for the first time.
We introduce a class of virtual knot invariants, called chord index invariants, and develop a general framework for lifting them to more refined, higher-order invariants. Applying this procedure to the affine index polynomial, we obtain the first-order refinement, termed the R invariant, and the second-order refinement, the S invariant. We collectively refer to these as chord degree invariants. Their fundamental properties are analyzed, and potential applications are explored, including estimates of the minimal crossing number and the Gordian distance.
Consider the knots obtained from torus knots by inserting some number of full twists along two adjacent strands. We determine their knot types.
In this paper, we define a homotopy (we call it sq-homotopy) associated to the square product of two hypergraphs and provide a necessary condition for two hypergraph homomorphisms being homotopic defined by Yau et al. Then we prove that the sq-homotopy could be characterized by properties of an exponential hypergraph of type II. Finally, we define a Hom construction, and investigate a connection between the sq-homotopy and the Hom construction from the view of topology.
In this paper, we introduce the notion of symmetric biquandles and establish a shadow symmetric biquandle (co)homology theory. This framework provides shadow symmetric biquandle cocycle invariants for unoriented links in three-dimensional space and unoriented surface-links in four-dimensional space, represented by broken surface diagrams. We further interpret these invariants via marked graph diagrams (ch-diagrams) and examine their properties with illustrative computations. In addition, we classify all symmetric non-quandle biquandles together with corresponding good involutions of orders 2, 3, and 4 up to isomorphism.
This note is in two parts. The first part contains proofs of four elementary criteria for two permutations in the symmetric group S-n to be conjugate in S-n. Most readers know the first criterion but perhaps fewer are familiar with the remaining three. The second part contains a discussion of the problem of deciding whether two ordered pairs of permutations are conjugate by an element in S-n. In sharp contrast to the situation in part one, here no elementary criteria are known. However, pairs of permutations give rise to dessins d'enfants and two ordered pairs of permutations are conjugate precisely when the corresponding dessins are isomorphic. We give an expository discussion, with examples and references but no proofs, of some of this material; we hope our discussion will help readers who are encountering this known material for the first time.
In this paper, we study nilpotent p-localizations of knot groups using the symplectic automorphism groups of free nilpotent groups. We show that any function on the set of conjugacy classes in the corresponding outer automorphism group defines a knot invariant. Furthermore, we study these automorphism groups and compute several invariants arising from this construction.
In recent years, numerous polynomial invariants of knotoids have been constructed, some of which are defined with the signs of the crossings. In this paper, the coloring-allowed invariants of planar knotoids, which is a class of planar knotoid invariants defined with the coloring number are introduced. We demonstrate several basic properties of the coloring number and some examples of coloring-allowed invariants, among which the 4-phases functions are discussed in detail, including their invariance and properties.
A non-self OU sequence is a cyclic sequence of crossing information of non-self crossings that is obtained by traversing a knot component of an oriented link diagram. In this paper, we investigate what information can be derived from non-self OU sequences, and we completely characterize pairs of non-self OU sequences of diagrams of two-component links. We also characterize the pairs for specific prime links with crossing number up to five.
In this paper, we introduce a new equivalence relation, named R-equivalence relation, on the set of colorings of an oriented knot diagram by a quandle. We determine the R-equivalence classes of colorings of a diagram of a torus knot by a quandle, called Rot E-2, under a certain condition.
We prove a codimension-1 spun embedding result for Seifert 3-manifolds in a spin 4-manifold. This generalizes a previous result for lens spaces [S. Lawande and K. Saha, Twist maps and codimension-1 spun embeddings, preprint (2025), arXiv.2509.06168]. In particular, we provide sufficient conditions on the Seifert invariants of a Seifert 3-manifold to admit a spun embedding into a connected sum of copies of S-2 x S-2. We also give similar results for double branched covers over S-3 along infinitely many pretzel links and some Brieskorn homology 3-spheres, providing explicit spun embeddings in connected sums of S-2 x S(2)s and in the spin of lens spaces.
By extending the quiver state model that calculates the generalized Alexander polynomial defined by Kauffman and Radford, we introduce a new state model via the parity for crossings and define a polynomial called the even generalized Alexander polynomial in the paper. We prove that this polynomial is an invariant of virtual knots and show that there is a specific virtual knot that can be distinguished from the trivial knot by the polynomial. We introduce other generalized polynomial invariants via parity.
Blair-Campisi-Taylor-Tomova [Blair, Campisi, Taylor and Tomova, Kirby-Thompson distance for trisections of knotted surfaces, J. Lond. Math. Soc. (2) 105(2) (2022) 765-793.] defined the L-invariant L(F) of a knotted surface F, using pants complexes of trisection surfaces of bridge trisections of F. After that, Aranda-Pongtanapaisan-Zhang [Aranda, Pongtanapaisan and Zhang, Bounds for Kirby-Thompson invariants of knotted surfaces, Geom. Dedicata 217(6) (2023) 30] introduced the L-& lowast;-invariant L-& lowast;(F) using dual curve complexes instead of pants complexes. In this paper, we determine both the L-invariant and the L-& lowast;-invariant of any finite distant sum of standard surfaces, and this is the first example of knotted surfaces whose bridge numbers and these invariants can be arbitrarily large.
